What does “complete the square” mean?

Completing the square rewrites a quadratic into a perfect square trinomial.

This method always works.

Steps

1. Move the constant to the other side

2. Take half of the $x$-coefficient

3. Square it

4. Add it to both sides

5. Factor and solve

Example

x^2 + 4x = 5

Half of 4 is 2 → 2^2 = 4

x^2 + 4x + 4 = 9

(x + 2)^2 = 9

 x=−2±3